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My presentation at the IEEE - Internationa Reliability Physics Symposium 2007
Citation preview
MODELING OF INTERCONNECT DIELECTRIC LIFETIME UNDER STRESS
CONDITIONS AND NEW EXTRAPOLATION METHODOLOGIES
FOR TIME-DEPENDENTDIELECTRIC BREAKDOWN
Gaddi S. Haase, Joe W. McPhersonTexas Instruments, Dallas, TX, USA
TEXAS INSTRUMENTS
4E.2
2
• The actual minimum line-to-line spacing is dropping under 60nm for the new technology nodes.
• The traditional methodology for BEOL low-k dielectric lifetime predictions (E-model + Weibull statistics) might seem overly conservative, and might restrict technology scaling.
• Do we have enough data to support more lenient models for extrapolating test data to predict product lifetime?
• Is there any other source of over-conservatism that can be readily removed?
Purpose
3
• Two critically important parts of reliability assessments.
• Physical models for dielectric degradation as a function of E-field.
• Statistical Modeling: – Is Weibull good for interconnect dielectrics?
– Actual line-to-line spacing distributions
– Lifetime simulations using within-DUT variations
– Lifetime simulations including DUT to DUT variations.
– Are we using the correct Weibull parameters?
– A couple methodologies for extracting the true shape parameter β
• Conclusions
Outline
4
Ch
arac
teri
stic
life
tim
e (u
sua
lly
on
a l
og
sca
le)
Stress
The two parts of reliability assessments
1.What is the functional dependence of the lifetime on the stress? •Can the lifetime be extrapolated to use conditions?
Example: pdf of the Weibull
distribution for differentβ
values
Time
Prob
abili
ty D
ensi
ty
2. What statistical function describes dielectric failure?•Extract lifetime at cumulative failure probability <0.1%, while testing small DUT samples
•Area scaling
5
1.E-01
1.E+01
1.E+03
1.E+05
1.E+07
1.E+09
1.E+11
1.E+13
1.E+15
1.E+17
1.E+19
1.E+21
1.E+23
1.E+25
0 1 2 3 4 5 6
Electric Field (MV/cm)
Lif
etim
e
E model
SQRT(E) model
1/E model
Power-law
Usag
e fieldL
ifet
ime
(Lo
g s
cale
)Accelerated test regime
Extrapolating interconnect dielectric accelerated test data
• The E-model is generally the most conservative.
• The E-model is backed by a plausible physical model.
• Other models either lack viable physical basis for BEOL-dielectrics, or have no experimental verification.
• Uncertainty in E during test, and limited test time make model exploration & verification difficult.
• Hence, at this point, there is no safe, verified alternative to the E-model.
6
( )[ ] ( )[ ] NN tFtF −=− 11
Area scaling requires that:
( )
−−=
β
αt
tF exp1
Weibull distribution
Is Weibull statistics viable for BEOL dielectric BD?
F(t) = cum. failure probability at ≤ t
N= area factor. Correct ONLY if the cap dielectric properties & thickness are homogeneous
A tw
o param
. solution
-3.5
-2.5
-1.5
-0.5
0.5
1.5
0 1 10 100 1000 10000 100000Time to Breakdown (s)
ln[-
ln(1
-F)]
40 V
45 V
50 V
Log. (45V)
Linearized Weibull Plot
β= ~0.9But trying to fit TDDB data to a Weibull distribution…
• The fit is often poor
• The slope, β, appears <1
7
The actual line-to-line spacing distribution in a 65nm technology comb-comb test structure
The observed spacing distributions, from many cross-sectional electron micrographs for each test structure, were fitted with asymmetric normal distributions (solid curves) for ease of use in the simulation.
M1
M2
0
0.2
0.4
0.6
0.8
1
60 70 80 90Spacing at the top of the lines (nm)
Cu
m.
Pro
bab
ilit
y
Center
Edge
M1
8
TDDB Simulation: Within DUT variation only• All DUTs in the simulated test have the same spacing distribution.• In this example: All DUTs have the “Center” spacing.• Divide the test structure into bins i with spacing Si
• Use the E-model to correlate spacing (Si) with Field (Ei) and with characteristic lifetime ( tbd,63%, i ) :
0.0E+00
5.0E-04
1.0E-03
1.5E-03
2.0E-03
2.5E-03
60 65 70 75 80 85 90
Line-to-Line spacing (nm)
Nor
mal
ized
bin
pro
babi
lity
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
Ass
ocia
ted
tbd(
63%
) (s
)
Spacing distribution(Normal)
The associatedtbd(63%) at 50 V
Line-to-line spacing (nm)
CENTER
Nor
amal
ized
bin
pro
babi
lity
tbd, 63%,i (s)
Acceleration factor γ=4.0 cm/MV
50V
⋅−⋅= V
sAt
iibd
γexp%,63,
9
• All DUTs in the simulated test have the same spacing distribution.• In this example: All DUTs have the “Center” spacing.• The probability to find spacing bin i is
The “real” Weibull β
was taken as 2.0
The “experimentally observed” β is <1.7 at 50 V stress
y = 1.69Ln(x) + 0.96
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.0 0.1 1.0 10.0
Lifetime (s)
Cu
m. P
rob
ab
ility
-8
-6
-4
-2
0
2
4
ln[-
ln(1
-F)]
F vs. Lifetime at 50 V
ln[-ln(1-F)] vs. lifetime at 50 V
Linearized Weibull Plot fit
Lifetime (s)
CENTER50 VC
um fa
ilure
pro
b.
Ln[-ln(1-F)]
isp
TDDB Simulation: Within DUT variation only
( ) ( )[ ]∏ −−=i
pbdibd
istFtF 11
• Each bin i, with a constant spacing Si, has its own Weibull-ditribution Fi(tbd) with the associated tbd,I,63% , but the same β.
10
• The wafer is divided to several zones (locations), each having a different spacing-distribution and a lifetime-distribution Fzone(tbd)
• The occurrence of a tested DUT in each zone is: Pzone
• The total observed cum. failure probability would be:
Adding the DUT-to-DUT (across wafer) variations:
Simulated TDDB with two distinct spacing-distributions in two wafer area-loactions which are equally represented in the tested site map.
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05
Time to breakdown (s)
ln[-
ln(1
-F)]
50 V stress45 V stress40 V stress35 V stress
Note: The lifetime of DUTs from different zones are independent.
Simulated, using only the “center” and “edge” zones with equal contribution (Pzone=0.5)
( ) ( )∑ ⋅=zone
bdzonezonebd tFPtF
11
Simulated apparent TDDB
Using DUTs from three spacing-distribution zones. Simulated TDDB with three distinct spacing-distributions in three wafer area-loactions which are equally represented in the tested site map.
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05
Time to breakdown (s)
ln[-
ln(1
-F)]
50 V stress45 V stress40 V stress35 V stress
However, the “true” β used in this
simulation was 2.0
The “apparent” β is only ~1
Simulated data
Note: The true β is dictated by the actual material variation
12
Simulated TDDB with three distinct spacing-distributions in three wafer area-loactions which are equally represented in the tested site map.
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
1.E-03 1.E-01 1.E+01 1.E+03 1.E+05 1.E+07 1.E+09 1.E+11 1.E+13
Time to breakdown (s)
ln[-
ln(1
-F)]
50 V stress45 V stress40 V stress35 V stress1.5 V stress
At low voltage, β appears
again as 2.0
Extending the simulation to operating voltage :
Simulated data
At operating voltages, the spacing variations do not affect the lifetime uncertainty as much as at test voltages !
The “true” β value can be used for product reliability assessment
13
-5-4-3-2-10123
0.001 0.01 0.1 1 10
Time to Breakdown (s)
ln[-
ln(1
-F)]
Area x104 x103x102 x10 x1• Area scaling tests Simulations using:
( ) ( )[ ]NN tFtF −−= 11
•Center zone sites only•Used “real” β=2.0
Simulated data
Experimental extraction of the correct β
50VCenter
β1
,%63
%63 Nt
t
N
=
“Observed” β is -1/slope ≈1.93
• Extract β from tbd,63 vs. area:Slope = -0.52
-6
-5
-4
-3
-2
-1
0
1 10 100 1000 10000
Area factor
ln(
t 63
%/s
ec
)
Center50V stress
=N
14
• Area scaling tests. Simulations using:
( ) ( )[ ]NN tFtF −−= 11
•All three zones•Used “real” β=2.0
Simulated data
50VAll three zones
slope = -0.51
-5
-4
-3
-2
-1
0
1
1 100 10000
Area factor
ln(
t 63
%/s
ec
)
All zones50V stress
“Observed” β is -1/slope ≈1.96
• Extract β from tbd,63 vs. area:
=N
-5
-3
-1
1
3
0.001 0.01 0.1 1 10
Time to Breakdown (s)ln
[-ln
(1-F
)]
Area x104 x103 x102 x10 x1
Experimental extraction of the correct β
Simulated data
15
Using V-ramp to breakdown for test structures with various line-length allows more sampling:
TDDB tests are slow. A faster way to evaluate β:
100mV/170ms
0.E+00
1.E-03
2.E-03
3.E-03
4.E-03
5.E-03
6.E-03
50 60 70 80 90
Line-line spacing (nm)
Pro
bab
ilit
y to
fi
nd
sp
acin
g
0
10
20
30
40
50
60
70
Vb
d,6
3%
testi
testiiibd V
t
VtsV +
=
,0
%63,%63,,
)(ln
γ
Vs
i
ie
t∆⋅−
−
∆= γτ
1,0
Every spacing bin Si
has its own Vbd distribution with a
characteristic Vbd,63%,i
A. Berman, 1981
0.001
0.002
0.003
0.004
0.005
0.006
16
100mV/170msAll three zones
-4
-3
-2
-1
0
1
35 40 45 50 55Vbd (Volts)
ln[-
ln(1
-F)]
Area X100 X10 X1
Simulated data
( )( )
βγ ⋅=%63
%,63,
%63,
exp
exp s
Nbd
bd NV
V
-7-6-5-4-3-2-10123
35 40 45 50 55Vbd (Volts)
ln[-
ln(1
-F)]
100mV/170ms
100mV/1770ms
All three zones 4.437V
4.201V
3.982V
( ) ( )[ ]
∆
∆−=
2
1
12%63
ln
%63,%63,
ττγ
RVRVs bdbd
Simulated data
Using V-ramp to breakdown at two ramp rates, si/γ can be extracted at F=63%, and used to derive β from Vbd,63% vs. area curves.
Evaluating the true β with V-ramp tests
17
Conclusions• Simulations of lifetime distributions were performed using:
– Actual line-to-line spacing distributions in a test structure– Actual across-wafer spacing variations– Weibull statistics for each spacing bin– E-model (the most conservative option)
• The resulting “observed” accelerated test lifetime does not appear Weibull distributed.
• A Weibull-plot fit for a typical sample size shows a substantially lower “observed” β (Weibull shape parameter).
• Simulations at low (usage) voltage show that only the true β matters! (which depends only on actual material variations)
• Using the true β buys us the extra lifetime margin that we need (until we have other validated physical degradation models).
• The true β can be approximated from line/area-scaled testing.
• A faster true-β extraction can be done using two-ramp rate V-ramp to breakdown on multiple length test structures.
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